Integrand size = 15, antiderivative size = 27 \[ \int \frac {e^{2 x}}{\sqrt [3]{1+e^x}} \, dx=-\frac {3}{2} \left (1+e^x\right )^{2/3}+\frac {3}{5} \left (1+e^x\right )^{5/3} \]
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Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2280, 45} \[ \int \frac {e^{2 x}}{\sqrt [3]{1+e^x}} \, dx=\frac {3}{5} \left (e^x+1\right )^{5/3}-\frac {3}{2} \left (e^x+1\right )^{2/3} \]
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Rule 45
Rule 2280
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {x}{\sqrt [3]{1+x}} \, dx,x,e^x\right ) \\ & = \text {Subst}\left (\int \left (-\frac {1}{\sqrt [3]{1+x}}+(1+x)^{2/3}\right ) \, dx,x,e^x\right ) \\ & = -\frac {3}{2} \left (1+e^x\right )^{2/3}+\frac {3}{5} \left (1+e^x\right )^{5/3} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {e^{2 x}}{\sqrt [3]{1+e^x}} \, dx=\frac {3}{10} \left (1+e^x\right )^{2/3} \left (-5+2 \left (1+e^x\right )\right ) \]
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Time = 0.04 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.56
method | result | size |
risch | \(\frac {3 \left (-3+2 \,{\mathrm e}^{x}\right ) \left (1+{\mathrm e}^{x}\right )^{\frac {2}{3}}}{10}\) | \(15\) |
default | \(-\frac {3 \left (1+{\mathrm e}^{x}\right )^{\frac {2}{3}}}{2}+\frac {3 \left (1+{\mathrm e}^{x}\right )^{\frac {5}{3}}}{5}\) | \(18\) |
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none
Time = 0.28 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.52 \[ \int \frac {e^{2 x}}{\sqrt [3]{1+e^x}} \, dx=\frac {3}{10} \, {\left (2 \, e^{x} - 3\right )} {\left (e^{x} + 1\right )}^{\frac {2}{3}} \]
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Time = 0.38 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {e^{2 x}}{\sqrt [3]{1+e^x}} \, dx=\frac {3 \left (e^{x} + 1\right )^{\frac {5}{3}}}{5} - \frac {3 \left (e^{x} + 1\right )^{\frac {2}{3}}}{2} \]
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none
Time = 0.20 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.63 \[ \int \frac {e^{2 x}}{\sqrt [3]{1+e^x}} \, dx=\frac {3}{5} \, {\left (e^{x} + 1\right )}^{\frac {5}{3}} - \frac {3}{2} \, {\left (e^{x} + 1\right )}^{\frac {2}{3}} \]
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none
Time = 0.30 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.63 \[ \int \frac {e^{2 x}}{\sqrt [3]{1+e^x}} \, dx=\frac {3}{5} \, {\left (e^{x} + 1\right )}^{\frac {5}{3}} - \frac {3}{2} \, {\left (e^{x} + 1\right )}^{\frac {2}{3}} \]
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Time = 0.06 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.52 \[ \int \frac {e^{2 x}}{\sqrt [3]{1+e^x}} \, dx=\frac {3\,{\left ({\mathrm {e}}^x+1\right )}^{2/3}\,\left (2\,{\mathrm {e}}^x-3\right )}{10} \]
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